SomepropertiesofFibonacciandChebyshev polynomials

Li Advances in Difference Equations (2015) 2015:118 DOI 10.1186/s13662-015-0420-z

R E S E A R C H Open Access

Some properties of Fibonacci and ChebyshevpolynomialsYang Li*

*Correspondence:[email protected] of Mathematics,Northwest University, Xi’an, Shaanxi,P.R. China

AbstractIn this paper, we study the properties of Chebyshev polynomials of the first andsecond kind and those of Fibonacci polynomials and use an elementary method togive Chebyshev polynomials of the first and second kind in terms of Fibonaccipolynomials and vice versa. Finally, we get some identities involving the Fibonaccinumbers and the Lucas numbers.

Keywords: elementary method; Chebyshev polynomials; Fibonacci polynomials;Fibonacci numbers; Lucas numbers

1 IntroductionAs we know, the Chebyshev polynomials and Fibonacci polynomials are usually definedas follows: Chebyshev polynomials of the first kind are Tn+(x) = xTn+(x) – Tn(x), n ≥ ,with the initial values T(x) = , T(x) = x; Chebyshev polynomials of the second kind areUn+(x) = xUn+(x) – Un(x), n ≥ , with the initial values U(x) = , U(x) = x; Fibonaccipolynomials are Fn+(x) = xFn+(x) + Fn(x), n ≥ , with the initial values F(x) = , F(x) = .From the second-order linear recurrence sequences, we have

Tn(x) =[(

x +√

x – )n +

(x –

√x –

)n],

Un(x) =

√

x –

[(x +

√x –

)n+ –(x –

√x –

)n+],

Fn(x) =

n√

x +

[(x +

√x +

)n –(x –

√x +

)n].

These polynomials play a very important role in the study of the theory and applicationof mathematics, and they are closely related to the famous Fibonacci numbers {Fn} andLucas numbers {Ln} which are defined by the second-order linear recurrence sequences

Fn+ = Fn+ + Fn,

Ln+ = Ln+ + Ln,

where n ≥ , F = , F = , L = and L = . Therefore, many authors have investigatedthese polynomials and got many properties and corollaries. For example, Zhang [] uses

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Li Advances in Difference Equations (2015) 2015:118 Page 2 of 12

the Chebyshev polynomials and has obtained the general formulas involving Fn and Ln,

∑

a+a+···+ak+=nFm(a+) · Fm(a+) · · ·Fm(ak++)

= (–i)mn Fk+m

k · k!U (k)

n+k

(imLm

),

∑

a+a+···+ak+=n+k+

Lm(a+) · Lm(a+) · · ·Lm(ak++)

= (–i)m(n+k+) k!

k+∑

h=

(im+Lm

)h (k + )!h!(k + – h)!

U (k)n+k+–h

(imLm

),

where k, m are any positive integers, a, a, . . . , ak+ are nonnegative integers and i is thesquare root of –. Falcón and Plaza [, ] presented many formulas about Fibonacci poly-nomials. This fact allowed them to present a family of integer sequences in a new anddirect way. Zhang [] also used the Chebyshev polynomials to solve some calculating prob-lems of the general summations. Wu and Yang [] studied Chebyshev polynomials and gota lot of properties.

In this paper, we combine Sergio Falcón and Wenpeng Zhang’s ideas. Then we obtainthe following theorems and corollaries. These results strengthen the connections of twokinds of polynomials. They are also helpful in dealing with some calculating problems ofthe general summations or studying some integer sequences.

Theorem For any positive integer n, we have the identities

Tn(x) =n+∑

s=

n∑

k=

k · (sn – n)(n + k – )!Fs–(x)(–)s+n–(n – k)!(k + s)!(k – s + )!

,

Tn–(x) =n∑

s=

n∑

k=

k–(ns – s)(n + k – )!Fs(x)(–)s+n(n – k)!(k + s)!(k – s)!

.

Theorem For any positive integer n, we have

Un(x) =n+∑

s=

n∑

k=

k–( – s)(n + k)!Fs–(x)(–)s+n(n – k)!(k + s)!(k – s + )!

,

Un–(x) =n∑

s=

n–∑

k=

k+s(n + k – )!(–)s+nFs(x)(n – k + )!(k + s + )!(k – s + )!

.

Theorem For any positive integer n, we have the following forms:

Fn–(x) =n–∑

k=

n–∑

j=

j+–n(k + )(n – j – )!j!(n – k – j – )!(n + k – j)!

Uk(x),

Fn(x) =n–∑

k=

n–∑

j=

j–n+k(n – j – )!j!(n – k – j)!(n + k – j)!

Uk(x).


Theorem For any positive integer n, we have the following forms:

Fn(x) =n∑

k=

n–∑

j=

j+–n(n – j – )!Tk–(x)j!(n + k – j – )!(n – k – j)!

,

Fn–(x) =n–∑

k=

n–∑

j=

j+–n(n – j – )!Tk(x)j!(n + k – j – )!(n – k – j – )!

+n–∑

j=

j+–n(n – j – )!j!(n – j – )!(n – j – )!

.

Corollary For any positive integer n, we have the following identities:

Fn–

(imLm

)=

n–∑

k=

n–∑

j=

j+–n(k + )(n – j – )!Fm(k+)

j!(n – k – j – )!(n + k – j)!(–)kmFm,

Fn

(imLm

)=

n–∑

k=

n–∑

j=

j–n+k(n – j – )!(–i)Fkm

(–)kmj!(n – k – j)!(n + k – j)!Fm.

Corollary For any positive integer n, we have the following identities:

Fn

(imLm

)=

n∑

k=

n–∑

j=

j+–n(n – j – )!(–)km(–i)Lkm–m

j!(n + k – j – )!(n – k – j)!,

Fn–

(imLm

)=

n–∑

k=

n–∑

j=

j+–n(n – j – )!(–)kmLkm

j!(n + k – j – )!(n – k – j – )!

+n–∑

j=

j+–n(n – j – )!j!(n – j – )!(n – j – )!

.

2 Some lemmasLemma For any nonnegative integers m and n, we have these identities

∫

–

Tm(x)Tn(x)√ – x

dx =

⎧⎪⎨

⎪⎩

, m �= n,π , m = n > ,π , m = n = ;

()

∫

–Um(x)Un(x)

√ – x dx =

{, m �= n,π , m = n;

()

Tn(cos θ ) = cos nθ ; ()

Un(cos θ ) =sin (n + )θ

sin θ. ()

Proof See reference []. �

Lemma For any positive integers m and n, we have these identities

Un

(i

)= inFn+,

Tn

(i

)=

in

Ln+,


Tn

(Tm

(i

))= imnLmn,

Un

(Tm

(i

))= imn Fm(n+)

Fm.


Lemma For any positive integer n, we have

Tn(Tm(x)

)= Tnm(x),

Un(Tm(x)

)=

Um(n+)–(x)Um–(x)

.


Lemma For any positive integer n, let

Fn(x) =+∞∑

k=

ankUk(x) ()

and

Fn(x) =

bnT(x) ++∞∑

k=

bnkTk(x), ()

then we can get

ank =

{∑� n– �

j=(k+)(n–j–)!

j!(n–k–j–)!!(n+k+–j)!! , k + n is odd,, otherwise,

()

bnk =

{∑� n– �

j=(n–j–)!

j!(n+k–j–)!!(n–k–j–)!! , k + n is odd,, otherwise.

()

Proof To begin with, we multiply√

– xUm(x) to both sides of (), then integrate it from– to , we can get the following identity by applying property ():

∫

–

√ – xFn(x)Um(x) dx =

∞∑

k=

∫

–ank

√ – xUm(x)Uk(x) dx =

π

anm,

and then we have

anm =π

∫

–

√ – xFn(x)Um(x) dx.

From reference [] we know

Fn(x) =� n–

�∑

j=

(n – j –

j

)xn–j–, ()


where n ≥ . We define

wnk =π

∫ π

cosn θ sin(k + )θ sin θ dθ .


wnk =

{(k+)n!

(n+k+)!!(n–k)!! , k + n is even and n ≥ k,, otherwise,

()

where n and k are any nonnegative integers. Let x = cos θ , then we can get the followingidentity by applying property () and property ():

anm =π

∫ π

Fn(cos θ ) sin(m + )θ sin θ dθ

=π

� n– �∑

j=

(n – j –

j

)∫ π

cosn–j– θ sin(m + )θ sin θ dθ

=� n–

�∑

j=

(n – j –

j

)wn–j–,m,

and then we have anm = if n + m is even. If n + m is odd, we have

anm =� n–

�∑

j=

(n – j –

j

)· (m + )(n – j – )!

(n – m – j – )!!(n + m + – j)!!

=� n–

�∑

j=

(n – j – )!j!(n – j – )!

· (m + )(n – j – )!(n – m – j – )!!(n + m + – j)!!

=� n–

�∑

j=

(m + )(n – j – )!j!(n – m – j – )!!(n + m + – j)!!

.

We finish proving property ().In order to prove property (), we must multiply Tm(x)√

–x to both sides of (), then integrateit from – to , we can get the following identity by applying property ():

∫

–

Fn(x)Tm(x)√ – x

dx =∫

–

bnT(x)Tm(x)√

– xdx +

∞∑

k=

∫

–

bnkTm(x)Tn(x)√ – x

dx =π

bnm,

and then we have

bnm =π

∫

–

Fn(x)Tm(x)√ – x

dx.

We define

qnk =π

∫ π

cosn θ cos kθ dθ .



qnk =

{n!

(n+k)!!(n–k)!! , k + n is even and n ≥ k,, otherwise,

()

where n and k are any nonnegative integers. Let x = cos θ , then we can get the followingidentity by applying property () and property ():

bnm =π

∫ π

Tm(cos θ )Fn(cos θ )sin θ

sin θ dθ

=π

� n– �∑

j=

(n – j –

j

)∫ π

cosn–j– θ cos mθ dθ

=� n–

�∑

j=

(n – j –

j

)qn–j–,m,

and then we have bnm = if n + m is even. If n + m is odd, we have

bnm =� n–

�∑

j=

(n – j –

j

)· (n – j – )!

(n + m – j – )!!(n – m – j – )!!

=� n–

�∑

j=

(n – j – )!j!(n – j – )!

· (n – j – )!(n + m – j – )!!(n – m – j – )!!

=� n–

�∑

j=

(n – j – )!j!(n + m – j – )!!(n – m – j – )!!

.

This proves Lemma . �

Lemma For any positive integers m and n, we have the following identities:

Fn(i cos θ ) =in+ sin nθ

sin θ, ()

∫ i

–i

√x + Fm(x)Fn(x) dx =

{im–π , m = n > ,, otherwise.

()

Proof As we know,

Fn(x) =

n√

x +

[(x +

√x +

)n –(x –

√x +

)n].

Let x = i cos θ , then we have

Fn(i cos θ ) =

sin θ

[(i cos θ + sin θ )n – (i cos θ – sin θ )n]

=

sin θ

(ine–inθ – ineinθ

)


=in

sin θ(cos nθ – i sin nθ – cos nθ – i sin nθ )

=in+ sin nθ

sin θ.

This proves property (). Let x = i cos θ in the following identity:

A =∫ i

–i

√x + Fm(x)Fn(x) dx,

then we can get

A =∫ π

sin θFn(i cos θ )Fm(i cos θ )i sin θ dθ

=∫ π

i sin θ

in+ sin nθ

sin θ

im+ sin mθ

sin θdθ

= in+m–∫ π

sin nθ sin mθ dθ

= in+m–∫ π

cos (n – m)θ – cos (m + n)θ dθ .

Then we can get property (). This proves Lemma . �

Lemma For any positive integer n, we have

Tn(x) =n∑

k=

(–)n–k · k · nn + k

(n + k

k

)xk ,

Tn+(x) =n∑

k=

(–)n–k · k · (n + )n + k +

(n + k +

k +

)xk+,

Un(x) =n∑

k=

(–)n–k · k · (k + )n + k +

(n + k +

k +

)xk ,

Un+(x) =n∑

k=

(–)n–k · k+ · (k + )n + k +

(n + k +

k +

)xk+.

Proof From Theorem of reference [], we can get the following result easily:

Tn(x) =n∑

k=

(–)n–k[

k(

n + kn – k

)– k–

(n + k –

k –

)]xk

=n∑

k=

(–)n–k · k–[

(n + k)!(k)!(n – k)!

–(n + k – )!

(k – )!(n – k)!

]xk

=n∑

k=

(–)n–k · k n(n + k – )!(k)!(n – k)!

xk

=n∑

k=

(–)n–k · k · nn + k

(n + k

k

)xk .


From Theorem of reference [], we know

Tn+(x) =n∑

k=

(–)n–k[

k+(

n + k + n – k

)– k

(n + k

k

)]xk+.

In a similar way, we can get

Tn+(x) =n∑

k=

(–)n–k · k · (n + )n + k +

(n + k +

k +

)xk+.

We can get T ′n(x) = nUn–(x) easily from the definition of the Chebyshev polynomials. If

we derive both sides of the above properties, we will get

Un(x) =n∑

k=

(–)n–k · k · (k + )n + k +

(n + k +

k +

)xk ,

Un+(x) =n∑

k=

(–)n–k · k+ · (k + )n + k +

(n + k +

k +

)xk+.



Tn(x) =+∞∑

s=

cn,sFs(x) ()

and

Tn–(x) =+∞∑

s=

cn–,sFs(x), ()

then we can get

cn,s =

{∑nk=

k+sn·is+n+(n+k–)!(n–k)!(k+s+)!!(k–s+)!! , s is odd,

, otherwise,()

cn–,s =

{∑nk=

(ns–s)is+n(n+k–)!–k (n–k)!(k+s)!!(k–s)!! , s is even,

, otherwise.()

Proof At first, we multiply√

x + Fm(x) to both sides of (), then integrate it from –ito i. We can get the following identity by applying Lemma , where m is any positiveinteger:

∫ i

–i

√x + Fm(x)Tn(x) dx =

∞∑

s=

∫ i

–icn,s

√x + Fs(x)Fm(x) dx = im–πcn,m,

and then we have

cn,m =(–i)m–

π

∫ i

–i

√x + Fm(x)Tn(x) dx.


Let x = i cos θ , then we can get the following identity by applying Lemma and property():

cn,m =(–i)m–

π

∫ π

Tn(i cos θ )Fm(i cos θ )i sin θ dθ

=–im

π

∫ π

Tn(i cos θ ) · im+ sin mθ

sin θ· ( sin θ ) dθ

=im+

π

∫ π

Tn(i cos θ ) sin mθ sin θ dθ

=n∑

k=

k · n · im+n+

n + k

(n + k

k

)wk,m–,

so when m is even, we have cn,m = . When m is odd, we have

cn,m =n∑

k=

k · n · im+n+

n + k·(

n + kk

)· m · (k)!

(k + m + )!!(k – m + )!!

=n∑

k=

k · n · im+n+

n + k· (n + k)!

(k)!(n – k)!· m · (k)!

(k + m + )!!(k – m + )!!

=n∑

k=

k+ · mn · im+n+(n + k – )!(n – k)!(k + m + )!!(k – m + )!!

.

In a similar way, we can get the following result easily:

cn–,m =(–i)m–

π

∫ π

Tn–(i cos θ )Fm(i cos θ )i sin θ dθ

=n∑

k=

k– · (n – ) · im+n

n + k –

(n + k –

k –

)wk–,m–,

and we have cn–,m = if m is odd. If m is even, we have

cn–,m =n∑

k=

k– · (n – ) · im+n

n + k –

(n + k –

k –

)m(k – )!

(k + m)!!(k – m)!!

=n∑

k=

(nm – m)im+n(n + k – )!–k(n – k)!(k + m)!!(k – m)!!

.



Un(x) =+∞∑

s=

dn,sFs(x) ()

and

Un–(x) =+∞∑

s=

dn–,sFs(x), ()


then we can get

dn,s =

{∑nk=

k+·is+n+s(n+k)!(n–k)!(k+s+)!!(k–s+)!! , s is odd,

, otherwise,()

dn–,s =

{∑n–k=

k+·is+ns(n+k–)!(n–k–)!(k+s+)!!(k+–s)!! , s is even,

, otherwise.()

Proof At first, we multiply√

x + Fm(x) to both sides of (), then integrate it from –ito i, we can get the following identities by applying Lemma , where m is any positiveinteger:

∫ i

–i

√x + Fm(x)Un(x) dx =

∞∑

k=

∫ i

–idns

√x + Fs(x)Fm(x) dx = im–πdn,m,

then we have

dn,m =(–i)m–

π

∫ i

–i

√x + Fm(x)Un(x) dx.

Let x = i cos θ , then we can get the following identity by applying Lemma and property():

dn,m =im+

π

∫ π

Un(i cos θ )Fm(i cos θ )i sin θ dθ

=im+

π

∫ π

Un(i cos θ )

im+ sin mθ

sin θ( sin θ ) dθ

=im+

π

∫ π

Un(i cos θ ) sin mθ sin θ dθ

=n∑

k=

k · (k + ) · im++n

n + k +

(n + k +

k +

)wk,m–,

so we have dn,m = if m is even. If m is odd, we have

dn,m =n∑

k=

k · (k + ) · im++n

n + k +

(n + k +

k +

)· m · (k)!

(k + m + )!!(k – m + )!!

=n∑

k=

k · (k + ) · im++n

n + k + · (n + k + )!

(k + )!(n – k)!· m · (k)!

(k + m + )!!(k – m + )!!

=n∑

k=

k+ · im+n+m(n + k)!(n – k)!(k + m + )!!(k – m + )!!

.

In a similar way, we have

dn–,m =(–i)m–

π

∫ π

Un–(i cos θ )Fm(i cos θ )i sin θ dθ

=n–∑

k=

k+ · (k + ) · im+n

n + k +

(n + k + k +

)wk+,m–.


When m is odd, we have dn–,m = . When m is even, we have

dn–,m =n–∑

k=

k+ · (k + ) · im+n

n + k +

(n + k + k +

)m(k + )!

(k + m + )!!(k + – m)!!

=n–∑

k=

k+ · im+nm(n + k – )!(n – k – )!(k + m + )!!(k + – m)!!

.


3 Proof of the theorems and corollariesIn this section, we will prove our theorems and corollaries. First of all, we can prove all thetheorems from Lemma , Lemma and Lemma easily.

Proof of Corollary We can get the following properties from Lemma and Lemma byletting x = Tm(x) in Theorem :

Fn–(Tm(x)

)=

n–∑

k=

n–∑

j=

j+–n(k + )(n – j – )!j!(n – k – j – )!(n + k – j)!

Uk(Tm(x)

),

Fn(Tm(x)

)=

n–∑

k=

n–∑

j=

j–n+k(n – j – )!j!(n – k – j)!(n + k – j)!

Uk(Tm(x)

).

Then, taking x = i in the above identities, according to Lemma , we can get Corollary .

�

Proof of Corollary We can get the following properties from Lemma and Lemma byletting x = Tm(x) in Theorem :

Fn(Tm(x)

)=

n∑

k=

n–∑

j=

j+–n(n – j – )!Tk–(Tm(x))j!(n + k – j – )!(n – k – j)!

,

Fn–(Tm(x)

)=

n–∑

k=

n–∑

j=

j+–n(n – j – )!Tk(Tm(x))j!(n + k – j – )!(n – k – j – )!

+n–∑

j=

j+–n(n – j – )!j!(n – j – )!(n – j – )!

T(Tm(x)

).

Then, taking x = i in the above identities, according to Lemma , we can get Corollary .

�

Competing interestsThe author declares that there is no conflict of interests regarding the publication of the article.

AcknowledgementsThe author would like to thank the referees for their very helpful and detailed comments which have significantlyimproved the presentation of this paper.

Received: 28 November 2014 Accepted: 18 February 2015


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1005-1019 (2009)3. Falcón, S, Plaza, Á: The k-Fibonacci sequences and the Pascal 2-triangle. Chaos Solitons Fractals 38, 38-49 (2007)4. Wu, X, Yang, G: The general formula of Chebyshev polynomials. J. Wuhan Transp. Univ. 24, 573-576 (2000)5. Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions: With Formulas, Graphs and Mathematical Tables,

chap. 22, pp. 771-802. Dover New York (1965)6. Ma, R, Zhang, W: Several identities involving the Fibonacci numbers and Lucas numbers. Fibonacci Q. 5, 164-171

(2007)

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SomepropertiesofFibonacciandChebyshev polynomials